Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)

2015 ◽  
Vol 65 (5) ◽  
Author(s):  
Ali M. Akhmedov ◽  
Saad R. El-Shabrawy
Keyword(s):  
Continuous Spectrum ◽  
Point Spectrum ◽  
Band Matrices ◽  
Residual Spectrum ◽  
Band Matrix ◽  
The Matrix ◽  
Lower Triangular

AbstractLet Δa,b denote an infinite lower triangular double-band matrix. In this paper, the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the matrix Δ

scholarly journals Fine Spectra of Upper Triangular Triple-Band Matrices over the Sequence Space ()

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Ali Karaisa ◽  
Feyzi Başar
Keyword(s):  
Continuous Spectrum ◽  
Sequence Space ◽  
Point Spectrum ◽  
Band Matrices ◽  
Residual Spectrum ◽  
Approximate Point Spectrum ◽  
Fine Spectrum ◽  
The Matrix ◽  
Lower Triangular ◽  
Triple Band

The fine spectra of lower triangular triple-band matrices have been examined by several authors (e.g., Akhmedov (2006), Başar (2007), and Furken et al. (2010)). Here we determine the fine spectra of upper triangular triple-band matrices over the sequence space . The operator on sequence space on is defined by , where , with . In this paper we have obtained the results on the spectrum and point spectrum for the operator on the sequence space . Further, the results on continuous spectrum, residual spectrum, and fine spectrum of the operator on the sequence space are also derived. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator over the space and we give some applications.

scholarly journals On the fine spectrum of generalized upper triangular triple-band matrices (Δ2uvw)t over the sequence space l1

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1363-1373 ◽  
Author(s):  
Selma Altundağ ◽  
Merve Abay
Keyword(s):  
Sequence Space ◽  
Point Spectrum ◽  
Matrix Operator ◽  
Band Matrices ◽  
Approximate Point Spectrum ◽  
Band Matrix ◽  
Fine Spectrum ◽  
The Matrix ◽  
Triple Band

In this work, we determine the fine spectrum of the matrix operator (?2uvw)t which is defined generalized upper triangular triple band matrix on l1. Also, we give the approximate point spectrum, defect spectrum and compression spectrum of the matrix operator (?2uvw)t on l1.

scholarly journals Fine Spectra of Upper Triangular Double-Band Matrices over the Sequence Space ,

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Ali Karaisa
Keyword(s):  
Sequence Space ◽  
Point Spectrum ◽  
Real Numbers ◽  
Band Matrices ◽  
Approximate Point Spectrum ◽  
Band Matrix ◽  
Fine Spectrum ◽  
The Matrix ◽  
Convergent Sequences

The operator on sequence space on is defined , where , and and are two convergent sequences of nonzero real numbers satisfying certain conditions, where . The main purpose of this paper is to determine the fine spectrum with respect to the Goldberg's classification of the operator defined by a double sequential band matrix over the sequence space . Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator over the space .

scholarly journals Evidence of the Poisson/Gaudin–Mehta phase transition for band matrices on global scales

2018 ◽  
Vol 07 (02) ◽  
pp. 1850002
Author(s):  
Sheehan Olver ◽  
Andrew Swan
Keyword(s):  
Phase Transition ◽  
Boundary Conditions ◽  
Rate Of Convergence ◽  
Critical Behavior ◽  
Level Density ◽  
Boundary Effects ◽  
Fourth Moment ◽  
Band Matrices ◽  
Band Matrix ◽  
Symmetric Band

We prove that the Poisson/Gaudin–Mehta phase transition conjectured to occur when the bandwidth of an [Formula: see text] symmetric band matrix grows like [Formula: see text] is naturally observable in the rate of convergence of the level density to the Wigner semi-circle law. Specifically, we show for periodic and non-periodic band matrices the rate of convergence of the fourth moment of the level density is independent of the boundary conditions in the localized regime [Formula: see text] with a rate of [Formula: see text] for both cases, whereas in the delocalized regime [Formula: see text] where boundary effects become important, the rate of convergence for the two ensembles differs significantly, slowing to [Formula: see text] for non-periodic band matrices. Additionally, we examine the case of thick non-periodic band matrices [Formula: see text], showing that the fourth moment is maximally deviated from the Wigner semi-circle law when [Formula: see text], and provide numerical evidence that the eigenvector statistics also exhibit critical behavior at this point.

PERTURBATION THEORY FOR THE ALFVÉN WAVE

1995 ◽  
Vol 09 (22) ◽  
pp. 2857-2898 ◽  
Author(s):  
Z. YOSHIDA ◽  
S.M. MAHAJAN
Keyword(s):  
Magnetic Field ◽  
Perturbation Theory ◽  
Electron Beam ◽  
Continuous Spectrum ◽  
Point Spectrum ◽  
Alfven Wave ◽  
Wave Number ◽  
Ambient Magnetic Field ◽  
Field Lines ◽  
The Magnetic Field

The Alfvén wave is the dominant low frequency transverse mode of a magnetized plasma. The Alfvén wave propagates along the magnetic field, and displays a continuous spectrum even in a bounded plasma. This is essentially due to the degeneracy of the wave characteristics, i.e. the frequency (ω) is primarily determined by the wave number in the direction parallel to the ambient magnetic field (k||) and is independent of the perpendicular wavenumbers. The characteristics, that are the direction along which the wave energy propagates, are identical to the ambient magnetic field lines. Therefore, the spectral structure of the Alfvén wave has a close relationship with the geometric structure of the magnetic field lines. In an inhomogeneous plasma, the Alfvén resonance (ω−cAk||=0; cA is the phase velocity of the Alfvén wave) constitutes a singularity for the defining wave equation; this results in a singular eigenfunction corresponding to the continuous spectrum. The aim of this review is to present an overview of the perturbation theory for the Alfvén wave. Emphasis is placed on those perturbations of the continuous spectrum which lead to the creation of point spectra. Such qualitative changes in the spectrum are relevant to many plasma phenomena. The first category of perturbations consists of nonideal effects such as the finite conductivity, kinetic effects arising from the finite electron inertia, and finite gyroradius. These effects add singular perturbations to the mode equation, and modify the spectrum dramatically. These modification, viz. the conversion of the continuous to the point spectrum, can lead to interesting physical phenomenon. A case in point is that of an electron beam propagating in a plasma which Cherenkov emits a left-hand circularly polarized Alfvén wave. The helicity of the ambient magnetic field imparts a frequency shift to the eigenmodes changing the critical velocity for Cherenkov emission. It, then, becomes possible for a sub-Alfvénic electron beam to excite a nonsingular Alfvén wave corresponding to a point spectrum. The second category comprises of geometric perturbations associated with higher dimensional inhomogeneity of the ambient field. Forbidden bands occur when a periodic modulation is applied. In a chaotic magnetic field, the weak localization of the wave occurs, resulting in a point spectrum.

scholarly journals The point spectrum and residual spectrum of upper triangular operator matrices

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1759-1771
Author(s):  
Xiufeng Wu ◽  
Junjie Huang ◽  
Alatancang Chen
Keyword(s):  
Hilbert Spaces ◽  
Point Spectrum ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operator Matrices ◽  
Residual Spectrum ◽  
Infinite Dimensional ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient

The point and residual spectra of an operator are, respectively, split into 1,2-point spectrum and 1,2-residual spectrum, based on the denseness and closedness of its range. Let H,K be infinite dimensional complex separable Hilbert spaces and write MX = (AX0B) ? B(H?K). For given operators A ? B(H) and B ? B(K), the sets ? X?B(K,H) ?+,i(MX)(+ = p,r;i = 1,2), are characterized. Moreover, we obtain some necessary and sufficient condition such that ?*,i(MX) = ?*,i(A) ?*,i(B) (* = p,r;i = 1,2) for every X ? B(K,H).

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